Digital Signal Processing Reference
In-Depth Information
2.5.2 METHOD OF ANALYSIS OF LTI SYSTEMS
The output generated by a linear, time-invariant (LTI) may be computed by considering the input (a
discrete time sequence of numbers) to be a sequence of individual sample-weighted, time-offset unit
impulses. The response of any LTI system to a single unit impulse is referred to as its
Impulse Response
.
The net response of the LTI system to an input sequence of sample-weighted, time-offset unit impulses
is the superposition, in time offset- manner, of its individual responses to each sample-weighted unit
impulse. Each individual response of the LTI system to a given input sample is its impulse response
weighted by the given input sample, and offset in time according to the position of the input sample in
time.
2
2
0
0
−2
−2
0
50
100
0
50
100
(a) n
(b) n
4
4
2
2
0
0
−2
−2
−4
−4
0
50
100
0
50
100
(c) n
(d) n
5
2
0
0
−2
−5
0
50
100
0
50
100
(e) n
(f) n
5
50
0
0
−50
−5
0
50
100
0
50
100
(g) n
(h) n
Figure 2.19:
(a)
x
1
[
]
; (b)
y
1
[
]
; (c)
x
2
[
]
; (d)
y
2
[
]
; (e)
ax
1
[
]
(circles) and
bx
2
[
]
(stars); (f )
ay
1
[
]
n
n
n
n
n
n
n
+
by
2
[
n
]
; (g)
ax
1
[
n
]
+
bx
2
[
n
]
; (h)
NLS (ax
1
[
n
]
+
bx
2
[
n
]
)
.
Figure 2.20 depicts this process for the three-sample signal sequence [1,-0.5,0.75], and an LTI sys-
tem having the impulse response 0
.
7
n
, where
n
= 0:1:
∞
. We cannot, obviously, perform the superposition
for all
n
(i.e., an infinite number of values), so we illustrate the process for a few values of
n
.
The process above, summing delayed, sample-weighted versions of the impulse response to obtain
the net output, can be performed according to the following formula, where the two sequences involved
are denoted
h
[
n
]
and
x
[
n
]
:
∞
y
[
k
]=
x
[
n
]
h
[
k
−
n
]
(2.6)
n
=−∞
Equation (2.6) is called the
convolution formula
.
